{"id":"80fd070d-a777-469d-9e69-dec157eb731f","arxiv_id":"2412.10375","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Introduces rank-k, joint k, and joint C numerical ranges for nonnegative matrices in max algebra and proves their elementary properties.","lead":"This paper introduces new analogues of numerical ranges, rank-k numerical ranges, and joint numerical ranges for matrices whose operations use maximum instead of addition. It proves their basic algebraic properties and works explicit examples, including banded Toeplitz matrices.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the dominated-column step in Theorem 4.7(vii) is valid by a simple pigeonhole argument; remaining issues are presentational.","rationale":"The reader's weakest assumption was that Theorem 4.7(vii) relies on an unproven dominated-column lemma whose failure would collapse the inclusion W^{k+1}_max(A) ⊆ W^k_max(A). On inspection the lemma is true: for each of the m coordinates, at most one index can be the unique maximizer, so with k+1 > m indices there is an index that is never a unique maximizer. The inequality then holds, and removing that column yields a valid X ∈ X_{n×k} with the same coordinatewise maxima. Thus the identified concern does not land as a mathematical flaw. The paper does contain genuine presentational issues: the construction in Example 2.4 appears to have a typo in the displayed matrix (the entries do not satisfy X^t A X = λI_2 as written, though the claimed range [8,10] is correct and achievable by a corrected construction), and the proof of Theorem 4.7(vii) has an index typo. These deserve correction but do not undermine the central results, which include the Toeplitz characterization and the nesting properties. Since the reader's conditional verdict was based primarily on the dominated-column gap, and that gap is fillable, the verdict need not be changed; minor revisions are still warranted for the typos.","tokens_in":21683,"tokens_out":19901,"duration_ms":200437,"concrete_test":"Provide the missing pigeonhole justification in the proof of Theorem 4.7(vii) and verify the inclusion computationally for small cases: enumerate all X ∈ X_{n×(k+1)} for n=4, m=2, k=2 and random nonnegative 4×4 matrices A1, A2, then check that every computed W^3_max tuple lies in W^2_max; repeat for n=5, k=3, m=3. If any tuple fails, the inclusion as stated is false; otherwise the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's flagged concern is the assertion in Theorem 4.7(vii) that for k+1 > m there exists a column s with x_s^t A_j x_s ≤ ⊕_{i≠s} x_i^t A_j x_i for all j. This claim is actually true: for each coordinate j, at most one index i can be the unique maximizer of the (k+1)-tuple (x_i^t A_j x_i); any index tied at the maximum is not bad. Hence the set of 'bad' indices has size at most m, and since k+1 > m, some index s avoids being the unique maximizer in every coordinate. Removing that column preserves the isometry property because columns of X ∈ X_{n×(k+1)} have disjoint supports, so the remaining k columns still have max entry 1 and pairwise zero overlap; the max over the remaining columns equals the original λ_j coordinatewise. The proof should state this one-line pigeonhole argument, but its omission is not a correctness gap. The paper's other weaknesses are typos, such as the displayed X in Example 2.4 appearing to have misordered entries and the index in the last displayed line of Theorem 4.7(vii) being i≠s rather than i≠j, none of which affect the central claims. No load-bearing mathematical flaw was found.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces the max rank-k numerical range Λmax_k(A) for a nonnegative matrix A in the max algebra (Definition 2.1), as the set of λ ∈ R+ for which X^t ⊗ A ⊗ X = λ I_k for some max-isometry X ∈ X_{n×k}. It develops basic properties (Proposition 2.5), derives explicit characterizations for Toeplitz matrices (Theorem 2.7), and computes several examples. It then extends the joint numerical range to tuples of matrices, defining the max joint k-numerical range W^k_max(A) (Definition 4.1) and the max joint c- and C-numerical ranges (Definitions 5.1 and 5.9), with a number of algebraic properties (Theorems 4.7 and 5.12). The proofs largely follow from prior results in [15,16], and the main new technical ingredient is the nesting W^{k+1}_max(A) ⊆ W^k_max(A) under the condition m−1 < k < n.","tokens_in":22026,"tokens_out":20260,"duration_ms":185431,"significance":"If the results are accepted, the paper provides a coherent max-algebra analogue of higher-rank numerical ranges and their joint/C variants. The explicit computations for Toeplitz matrices (Theorem 2.7) and block-diagonal examples (2.4, 2.6) are useful for illustrating the non-classical behavior, e.g., Λmax_2(A) may be a non-degenerate interval even though the classical higher-rank ranges are intervals too. The paper is honest about an open problem on connectedness (Remark 4.5). It does not provide machine-checked proofs or numerical code; its strengths are the clear definitions and the explicit, checkable examples. The dependence on [15,16] for the description of unitary matrices as permutation matrices and for W^k_max characterization is explicit and not circular.","major_comments":[{"comment":"The proof asserts that since k+1 > m, there exists s such that x_s^t A_j x_s ≤ ⊕_{i≠s} x_i^t A_j x_i for all j. This is the key step in proving W^{k+1}_max(A) ⊆ W^k_max(A), and it is stated without justification. The statement is true: for each coordinate j, at most one index is the strict maximizer of (x_i^t A_j x_i)_{i=1}^{k+1}, so the set of indices that are strict maximizers in at least one coordinate has size at most m; since k+1 > m, some s is not a strict maximizer in any coordinate. The authors should include this argument, and also note that deleting column s from X preserves X ∈ X_{n×k}. As written, the proof is incomplete at a load-bearing point.","section":"Theorem 4.7(vii)"}],"minor_comments":[{"comment":"In the converse direction, the displayed matrix X appears to have misordered entries: with the entries as shown, the first column yields max(5, 8λ/10), not λ. The entry sqrt(λ/10) should be placed in the third coordinate, and the matrix convention (before/after transpose) should be clarified.","section":"Example 2.4"},{"comment":"In the last displayed line of the proof, the index in the deletion should be i ≠ s, not i ≠ j.","section":"Theorem 4.7(vii)"},{"comment":"The text 'there exist i ≤ k ≤ k and 1 ≤ j ≤ s' should read 'there exist i ∈ {1,...,k} and 1 ≤ j ≤ s', and 'Wmax(xj ⊗ jt_j)' should read 'Wmax(xj ⊗ y_j^t)'.","section":"Proposition 2.10"},{"comment":"The index condition should be 1 ≤ i_1 < ... < i_k ≤ n, not < n; and the equality W^k_max(A) = {max_i x_i y_i} is true only for k = n, not for general k.","section":"Remark 3.4"},{"comment":"In parts (i) and (ii), the formulas use ⊕ where the max-algebra product ⊗ is intended; for example, α_i ⊕ (⊕_{j=1}^n c_j) should be α_i ⊗ (⊕_{j=1}^n c_j), and c_1 ⊕ (⊕_{j=1}^n (A_i)_{jj}) should be c_1 ⊗ (⊕_{j=1}^n (A_i)_{jj}).","section":"Remark 5.4"},{"comment":"The displayed formula has an unmatched closing parenthesis after the m-th coordinate; add a closing parenthesis before the colon over the set.","section":"Proposition 5.2"},{"comment":"There are numerous typographical errors (e.g., 'Sloveniah' in the acknowledgments, 'Thaghizadeh' in reference [15], and inconsistent use of X_n×k vs X_{n×k}); these should be corrected in a final polish.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is within the scope of the journal and the main constructions are sound, but the proof of Theorem 4.7(vii) needs a small but essential justification. The paper also requires careful proofreading. I recommend requesting a revision rather than rejecting, since the technical gap is readily repairable and the remaining issues are expository."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the paper is sounder than the reader's report suggests. The one flagged gap in Theorem 4.7(vii) is not a gap. The dominated-column step follows from a one-line pigeonhole argument: for each j, at most one of the k+1 columns can be the unique maximizer of x_i^t A_j x_i, and with m such coordinates and k+1 > m, some column avoids being a unique maximizer in every coordinate, so dropping it preserves all λ_j. The authors should add that sentence, but the omission is presentational, not load-bearing.\n\nWhat is new: the max rank-k numerical range Λ^max_k(A), the max joint k-numerical range, and the max joint C-numerical range. These are natural extensions of the existing program from [15,16]. The Toeplitz theorem (Λ^max_2 = {a0}, empty for k ≥ 3) is a concrete new result with a real argument, and the worked examples check out. The reliance on the authors' earlier results is not circular; W^k_max and the permutation-matrix fact are independently published.\n\nSoft spots are mostly editorial. Proposition 2.10 contains 'i ≤ k ≤ k' and a garbled 'Wmax(xj ⊗ jt_j)'; the last line of Theorem 4.7(vii) writes i≠j where it should be i≠s; Example 2.4's displayed X is misordered. None of these affect the mathematics. The proof of Theorem 2.7 for k ≥ 3 is dense but appears sound.\n\nThe bigger picture: this is a niche contribution that extends a known toolkit rather than opening new technology. It does not resolve a long-open question. For the max-algebra numerical range community, though, it fills a real gap and supplies explicit formulas that were not there. A specialist will want the Toeplitz computation. The paper deserves a serious referee; with a pass fixing typos and adding the pigeonhole justification, it should be publishable.","headline":"The flagged gap in Theorem 4.7(vii) is a one-line pigeonhole argument, and the new max-algebra rank-k and joint ranges are a solid niche contribution.","tokens_in":22578,"tokens_out":4867,"would_cite":false,"duration_ms":45299,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["39B82","44B20","46C05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper transplants the classical rank-k numerical range into max algebra, proving a nested hierarchy that for Toeplitz matrices shrinks to the diagonal entry at rank two and disappears at higher ranks, and develops joint and…","keywords":["max algebra","rank-k numerical range","joint numerical ranges","max trace","Toeplitz matrices","nonnegative matrices","higher rank numerical radius","isometry in max algebra"],"falsifier":"Enumerate all $X\\in X_{4\\times 2}$ for the $4\\times4$ Toeplitz matrix with $a_0=1$, $a_1=2$, $a_{-1}=3$, $a_2=4$, $a_{-2}=5$: any solution of $X^t\\otimes A\\otimes X=\\lambda I_2$ with $\\lambda\\neq1$ disproves Theorem 2.7, as would any $X\\in X_{4\\times3}$ satisfying the rank-three equation. Separately, for $n=3$, $m=2$, $k=2$, an exhaustive search over all triples of disjoint-support unit columns and all diagonal pairs $(A_1,A_2)$ would settle whether every point of $W^3_{\\max}(\\mathcal{A})$ lies in $W^2_{\\max}(\\mathcal{A})$.","tokens_in":21483,"feed_emoji":"🧮","tokens_out":14771,"duration_ms":149889,"temperature":0.7,"pith_summary":"Max algebra replaces ordinary addition of nonnegative numbers by taking the larger of the two, while keeping ordinary multiplication; isometries in this setting are matrices whose columns have disjoint supports and max-norm one. The paper defines the max rank-$k$ numerical range $\\Lambda^{\\max}_k(A)$ as the set of nonnegative $\\lambda$ for which some max-isometry $X$ satisfies $X^t \\otimes A \\otimes X = \\lambda I_k$, and proves that these sets are nested, $W_{\\max}(A)=\\Lambda^{\\max}_1(A)\\supseteq\\Lambda^{\\max}_2(A)\\supseteq\\cdots\\supseteq\\Lambda^{\\max}_n(A)$. The sharpest result is an explicit computation for banded Toeplitz matrices: the rank-one range is an interval, the rank-two range is exactly the singleton $\\{a_0\\}$ containing the common diagonal entry, and every rank $k\\ge 3$ range is empty. For $m$-tuples of nonnegative matrices the paper introduces the max joint $k$-numerical range and the max joint $C$-numerical range, proving compactness, Lipschitz continuity, invariance under max-unitary conjugation, and an inclusion $W^{k+1}_{\\max}(\\mathcal{A})\\subseteq W^k_{\\max}(\\mathcal{A})$ when $k+1>m$. The upshot is a tropical analogue of higher-rank numerical ranges that is often finite or interval-valued, in contrast to the generally intractable complex case.","feed_headline":"Max-algebra rank-2 ranges shrink Toeplitz matrices to a single value","feed_subtitle":"For banded Toeplitz matrices, the max rank-2 range is exactly the diagonal entry; all higher ranks are empty.","key_machinery":"The central object is the max-isometry set $X_{n\\times k}=\\{X\\in M_{n\\times k}(\\mathbb{R}_+): X^t\\otimes X=I_k\\}$: its columns are nonnegative vectors of max-norm one with pairwise disjoint supports. The argument is carried by the equation $X^t\\otimes A\\otimes X=\\lambda I_k$, which forces each diagonal entry $x_j^t\\otimes A\\otimes x_j$ to equal $\\lambda$ and every off-diagonal block between two column supports to vanish; this support-disjointness is what makes the Toeplitz hierarchy collapse. For the joint range, the load-bearing mechanism is a pigeonhole step over $m$ coordinatewise maxima: if there are more columns than matrices, one column can be deleted without changing any of the $m$ max-trace entries, yielding the nesting inclusion.","core_discovery":"The paper's central claim is that the classical higher-rank numerical range has a faithful and computable analogue in max algebra. For an entrywise nonnegative matrix $A$, $\\Lambda^{\\max}_k(A)$ is defined by the equation $X^t \\otimes A \\otimes X = \\lambda I_k$ with $X$ ranging over max-isometries, and the paper establishes the basic structure of these sets: the $k=1$ case recovers $W_{\\max}(A)=[\\min_i a_{ii},\\max_{i,j}a_{ij}]$, the sets are nested in $k$, principal submatrices give subsets, and $\\Lambda^{\\max}_n(A)$ is nonempty exactly when $A$ is a scalar matrix. The flagship computation is Theorem 2.7: for a Toeplitz matrix with nonzero band entries and two zero corners, $\\Lambda^{\\max}_1(A)=[a_0,\\max_i a_i]$, $\\Lambda^{\\max}_2(A)=\\{a_0\\}$, and $\\Lambda^{\\max}_k(A)=\\emptyset$ for all $k\\ge 3$. The joint-range sections carry the same program to $m$-tuples, showing that $W^k_{\\max}(\\mathcal{A})$ is compact and locally Lipschitz and that it shrinks with $k$, with an inclusion $W^{k+1}_{\\max}(\\mathcal{A})\\subseteq W^k_{\\max}(\\mathcal{A})$ whenever $k+1>m$.","pith_inferences":["The paper leaves implicit that the support-disjointness mechanism gives a graph-theoretic reading: $\\Lambda^{\\max}_k(A)\\neq\\emptyset$ should force the existence of $k$ pairwise anticomplete vertex sets in the support digraph of $A$, making higher-rank ranges a combinatorial partition problem.","The pigeonhole deleted-column argument actually works for every $k\\ge m$, so the nesting in Theorem 4.7(vii) plausibly extends to all $k>m-1$; one testable consequence is that $W^k_{\\max}(\\mathcal{A})$ becomes constant for all $k\\ge m$.","Because every $X\\in U_n$ in max algebra is a permutation matrix, the max joint $C$-numerical range is a finite set; computing it is a bottleneck-assignment problem over permutations, so exact algorithms from max-plus optimization could evaluate it efficiently.","Connecting these ranges to max-plus spectral theory, the nesting may encode information about the tropical spectrum beyond the Perron root; a natural next step would be to check whether the support nodes selected by $\\Lambda^{\\max}_k(A)$ correspond to critical eigenvectors of matrix powers."],"forward_implications":["For every Toeplitz matrix of the stated banded form, membership in $\\Lambda^{\\max}_k(A)$ is decided by comparing $\\lambda$ to $a_0$: only $\\lambda=a_0$ works at rank two and nothing works at higher ranks.","The equality $\\Lambda^{\\max}_n(A)\\neq\\emptyset \\iff A=\\lambda I_n$ gives a max-algebra test for scalar matrices using the top of the rank hierarchy.","For an $m$-tuple, the inclusion $W^{k+1}_{\\max}(\\mathcal{A})\\subseteq W^k_{\\max}(\\mathcal{A})$ for $k+1>m$ implies the joint range stabilizes once the number of columns exceeds the number of matrices, and at $k=n$ it is the single point $(\\mathrm{tr}_\\otimes A_1,\\dots,\\mathrm{tr}_\\otimes A_m)$.","The rank-one bound of Proposition 2.10 yields an explicit upper bound on the max higher-rank numerical radius of a max-sum of rank-one matrices, so the radius can be estimated from the individual rank-one factors."],"supporting_citations":[{"why":"proves the interval formula $W_{\\max}(A)=[\\min_i a_{ii},\\max_{i,j}a_{ij}]$ that is the $k=1$ base of the hierarchy and of Theorem 2.7.","marker":"[14]"},{"why":"introduces the max joint numerical range and max $c$-numerical range that Sections 3--5 generalize to tuples and to rank $k$.","marker":"[15]"},{"why":"supplies the corrected interval characterization of $W^k_{\\max}(A)$ used in Theorem 3.2 and in the proof of Proposition 4.6.","marker":"[16]"},{"why":"provides the classical characterization $\\Lambda_k(A)=\\{\\lambda:X^*AX=\\lambda I_k\\}$ that motivates Definition 2.1.","marker":"[4]"},{"why":"defines the higher-rank numerical radius and product inequalities that Proposition 2.10 and the radius definition adapt to max algebra.","marker":"[5]"}],"fun_headline_variants":["Max rank-2 range collapses Toeplitz matrices to a point","Max algebra's higher ranks shrink ranges to points or empty","Nested max numerical ranges: from intervals to singletons","Max algebra introduces joint ranges for matrix tuples","Toeplitz max rank-2 range: exactly the diagonal entry"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof of the joint nesting inclusion rests on an unproved assertion in Theorem 4.7(vii): when there are more columns than matrices, some column of the isometry can be removed without changing any of the $m$ max-trace entries; if that assertion failed, the inclusion $W^{k+1}_{\\max}(\\mathcal{A})\\subseteq W^k_{\\max}(\\mathcal{A})$ would collapse.","fun_headline_variants_meta":{"raw":{"variants":["Max rank-2 range collapses Toeplitz matrices to a point","Max algebra's higher ranks shrink ranges to points or empty","Nested max numerical ranges: from intervals to singletons","Max algebra introduces joint ranges for matrix tuples","Toeplitz max rank-2 range: exactly the diagonal entry"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000461,"raw_usage":{"total_tokens":2301,"prompt_tokens":934,"completion_tokens":1367,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":550,"completion_tokens_details":{"reasoning_tokens":1283}},"tokens_in":550,"tokens_out":1367,"duration_ms":13901,"temperature":1.0,"reasoning_tokens":1283,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T16:23:35.287265+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Enumerate all $X\\in X_{4\\times 2}$ for the $4\\times4$ Toeplitz matrix with $a_0=1$, $a_1=2$, $a_{-1}=3$, $a_2=4$, $a_{-2}=5$: any solution of $X^t\\otimes A\\otimes X=\\lambda I_2$ with $\\lambda\\neq1$ disproves Theorem 2.7, as would any $X\\in X_{4\\times3}$ satisfying the rank-three equation. Separately, for $n=3$, $m=2$, $k=2$, an exhaustive search over all triples of disjoint-support unit columns and all diagonal pairs $(A_1,A_2)$ would settle whether every point of $W^3_{\\max}(\\mathcal{A})$ lies in $W^2_{\\max}(\\mathcal{A})$.","supporting_citations":[{"cited_title":"Tavakolipour and F","cited_arxiv_id":null,"evidence_quote":"proves the interval formula $W_{\\max}(A)=[\\min_i a_{ii},\\max_{i,j}a_{ij}]$ that is the $k=1$ base of the hierarchy and of Theorem 2.7."},{"cited_title":"Thaghizadeh, M","cited_arxiv_id":null,"evidence_quote":"introduces the max joint numerical range and max $c$-numerical range that Sections 3--5 generalize to tuples and to rank $k$."},{"cited_title":"Taghizadeh, M","cited_arxiv_id":null,"evidence_quote":"supplies the corrected interval characterization of $W^k_{\\max}(A)$ used in Theorem 3.2 and in the proof of Proposition 4.6."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the classical characterization $\\Lambda_k(A)=\\{\\lambda:X^*AX=\\lambda I_k\\}$ that motivates Definition 2.1."},{"cited_title":"Chiena, H-L","cited_arxiv_id":null,"evidence_quote":"defines the higher-rank numerical radius and product inequalities that Proposition 2.10 and the radius definition adapt to max algebra."}],"review_version":1}